2.5 Modeling of Extrusion Process
2.5.3 Upper Bound Method
tried by many researchers. The methods like upper bound analysis and finite element analysis were used to analyse the extrusion process.
experimental results. Sinha et al. [2009b] developed a simplified mathematical model and used upper bound method to obtain ram force in a multi-hole extrusion process. Theoretical and experimental ram forces were obtained for nine-hole, thirteen-hole and fifteen-hole dies with lead billets of same diameter and different lengths. The ram force was found to be the least for fifteen-hole die. Ram force also increases with increase in billet length.
Limited literature is available on the study of multi-hole extrusion process with upper bound method. However, sufficient number of research on single-hole extrusion has been carried out with upper bound method. Some relevant literatures are discussed here.
Kobayashi and Thomsen [1965] explained the method to find an admissible velocity field and stress field for axisymmetric forming problems. They also obtained an improvement of lower bound by modifying the admissible stress field for the frictionless extrusion of a bar through die with square corner. For the frictionless extrusion of a bar through tapered die, a lower bound solution was obtained for the extrusion pressure with reference to semi cone angle. Authors also suggested that if more definite information like redundant work of an extrusion problem is required, the general method of slip line solution may be used to get lower bound solutions.
For the extrusion of incompressible material, Chen and Ling [1968]
presented a method for selecting the admissible velocity fields. The upper bound method was used to find out mean extrusion pressure for the axisymmetric curved dies of cosine, elliptic and hyperbolic type. Osakada and Niimi [1975] suggested a generalized expression for radial flow field for extrusion through a conical die and obtained the extrusion pressure of a rigid-perfectly plastic material by using upper bound method. They also compared the extrusion pressure and hardness distribution obtained from the theoretical calculation with the hydrostatic extrusion of Al, Cu and Al-Cu composites.
Gunasekara et al. [1980] observed that manufacturing of shaped die is a difficult task in extrusion industry. They proposed upper bound solution for three dimensional metal flow for extrusion through dies having circular entry and regular polygonal exit profiles. The optimum die profiles were generated with the help of
computer to produce drawings of electrodes and then the dies were produced by electric discharge machining process. The reduced extrusion load was obtained from the shaped polished die compared to the flat faced die for same reduction ratio.
For non-axisymmetric extrusion process, Kiuchi [1988] developed a generalized mathematical equation of the kinematically admissible velocity field, which express the three dimensional steady flow of the workpiece. The extrusion pressure, the optimum die dimensions and the dimension of the extruded products were calculated. The developed velocity field was also applied to the analysis of eccentric backward extrusion for predicting the extrusion pressure and the geometry of the extruded products.
Yang et al. [1991] applied upper-bound method to determine the extrusion pressure for forward extrusion of composite rods through curved dies considering second order and third order flow functions. The effect of work-hardening was considered in the analysis. The experiments were carried out to verify the results obtained from upper bound method. The second-order flow function was in better agreement with the experimental observation than the third-order flow function both in extrusion load calculation and in deforming regions. The increase in semi-cone angle tends to minimize the extrusion power.
Altan et al. [1992] proposed a method of constructing kinematically admissible velocity field appropriate for axisymmetric extrusion and explained two deformation models (flow lines are straight line and flow lines are cubic) for extrusion. Reddy et al. [1995] proposed an upper-bound model combined with the slab method to predict the total extrusion power and die pressure distribution in axisymmetric extrusion process. The model proposed by the authors can be used for large class of die profiles in die design for extrusion and wire drawing. However, the authors have not verified their results with experiments with the proposed dies.
Celik and Chitkara [2000] investigated the off-centric extrusion of a square section from a round billet. To validate the results obtained from upper bound method, a number of the streamlined dies were designed by CAD/CAM package and manufactured by means of the EDM process. From the experiments, the authors observed that optimal die design depends significantly on the friction factor, which controls the curvature of the extruded products. Longer die lengths results in smaller
curvatures of the extruded products. The theory proposed by the authors gives estimation for the curvature of the exiting product.
Alexandrov et al. [2001] proposed a kinematically admissible velocity field based on simple radial flow field combined with asymptotic behavior of the velocity field for the analysis of axisymmetric direct and indirect extrusion. Authors also studied the influence of extrusion ratio on the shape of the dead zone and the extrusion pressure.
In general, the conventional extrusion is carried out by shear faced (flat faced) die. The use of sheared faced dies have many practical problems such as dead metal zone, more redundant work and above all the design of shear faced die is done based on experience and made by trial and error methods. But these methods are approximate and time-consuming methods. The profile of the extrusion dies is always an important parameter to optimize the extrusion pressure. Narayanasamy et al. [2006] proposed an upper bound analysis for the extrusion of circular section from circular billets by extruding through cosine die profile. They observed that under no friction condition, the cosine profile die consumes less extrusion pressure compared to straight converging and concave circular profile dies. It was further observed that the cosine profile die needs lower plastic deformation work. Die surface friction and total power consumption by using cosine profile dies was found to be less compared to straight converging die for all the values of relative die length.
The analyses on extrusion pressure by upper bound method have been extended to the extrusion of complex sections. Ajiboye and Adeyemi [2006, 2007]
carried out upper bound analysis on the effect of die land length on the extrusion pressure considering the ironing effect at die land region for different cross-sections of the extruded products. The effect of die land length on the extrusion pressure increases with increase in complexity of the cross-section of the extruded products.
Bakhshi-Jooybari et al. [2007] proposed a combined upper bound and slab method for estimating the deformation load for cold rod extrusion of aluminum and lead with an optimum curved die profile and optimum conical die. It was observed that optimum curved die requires lesser load than the optimum conical die. Experiments and finite element simulations were carried out for verifying the estimated extrusion
load. Malapani and Kumar [2007] carried out three dimensional analysis of tube extrusion process using upper bound method. The product profile, shape complexity factor, die and mandrel profile, friction, ram velocity and die length were considered as process parameter. It was observed that for any die profile, the average power varies linearly with friction factor.
Gordon et al. [2007] used upper bound modeling to determine adaptable die shapes that produce dies of specified criteria. The description for an extrusion die with a bearing length was developed for the use in the adaptable design method.
Optimal die shapes were determined based on the average effective strain criterion and the volumetric effective strain rate deviation criterion. The authors also proposed that an upper bound model can be used to analyze a multi-sectioned dies.
Altinbalik and Ayer [2008] proposed a new kinematically admissible velocity field to calculate the load requirements of extrusion of clover sections from cylindrical billets. The derived velocity field was found suitable to three dimensional extrusions of complex shapes. The quality of the extruded product was also investigated and it was observed that the value of deflection was quite small for all billet diameters which is in good agreement with the results obtained by Ajiboye [2007].
Huang et al. [2009] designed the streamline function considering the die profile and rigid plastic boundaries. The rigid plastic boundaries were the parametric functions with user defined variables which can be selected to minimize the power by upper bound method. The authors used four different die profiles, the cosine die with zero slope at entry and exit, the elliptic die with zero slope at entry, hyperbolic die with zero slope at exit and conical die with slope at entry and exit. The deformation pattern, extrusion pressure and the effective strain were calculated. The cosine die was found to consume lowest power where as the hyperbolic die consumes highest power.
The upper bound method has also been used to study the backward and radial extrusion processes. Plancak et al. [1992] investigated radial extrusion process with upper bound method and predicted the extrusion load. Bae and Yang [1992] applied an upper-bound method to determine the extrusion load and the deformed configuration for backward extrusion of internally elliptic-shaped tubes from round billets. Later, Bae and Yang [1993a, 1993b] presented a kinematically
admissible velocity field solution for the backward extrusion of tubes with internal trochoidal gear shape and rounded rectangular shape from the round shaped billets.
The authors also presented a kinematically admissible velocity field solution for the backward extrusion of internally circular shaped tube from regular polygonal and circular billets.
Non-axisymmetric combined forward and backward extrusion process was studied by Hwang et al. [2003] using the upper bound element technique (UBET).
The kinematically admissible velocity fields were proposed to determine the forming load, the extruded product length and the velocity distribution according to the stroke in the combined extrusion process. Ebrahimi et al. [2008] used analytical approach to obtain the extrusion pressure and the strain imposed through radial forward extrusion. They derived equations for extrusion pressure by determining the internal power and the power dissipated on all frictional and velocity discontinuity surfaces.
When a material is plastically deformed, a very large part of the work expended appears as heat energy. The temperature generated reduces the flow stress of the material, which consequently lowers the power required for deformation. A numerical method was developed by Ajiboye and Adeyemi [2008] to obtain the non-steady-state temperature distributions during forward extrusion process. The velocity, strain rates, and strain fields within the deformation zones during extrusion were obtained with upper bound method. Heat transfer equations were coupled with upper bound method to obtain internal heat generations.
The upper bound method also has been used to study the extrusion of porous metals. Oh and Phark [1987] developed an analytical model by using upper bound method for forward extrusion of porous metal through an axisymmetric square die.
The shape of the boundary of the dead metal zone (linear, bi-quadratic and hyperbolic) and the relative density of the product were optimally determined by minimizing the relative mean extrusion pressure. Experiments were also carried out to verify the results obtained from the developed model. From experiments, it was also observed that the densification does not occur in the deformation zone in the steady state condition. The analytical results of hyperbolic boundaries were in good agreement with the experimental results.
Use of upper bound method to solve many problems of extrusion process led much research activities aiming in finding the new types of velocity fields and optimization of power consumption. The application of upper bound method to solve three dimensional problems also has established a wider horizon for design of different die profiles and production of different complex extruded profiles. The researchers have given more emphasis on reduction of power consumption by using correct die. The upper bound method does not give clear idea about the quality and properties of the extruded products.